La geometrie rigide est devenue, au fil des ans, un outil indispensable dans un grand nombre de questions en geometrie arithmetique. Depuis ses premieres fondations, posees par J. Tate en 1961, la theorie s'est developpee dans des directions variees. Ce livre est le premier volume d'un traite qui expose un developpement systematique de la geometrie rigide suivant l'approche de M. Raynaud, basee sur les schemas formels a eclatements admissibles pres. Ce volume est consacre a la construction des espaces rigides dans une situation relative et a l'etude de leurs proprietes geometriques. L'accent est particulierement mis sur l'etude de la topologie admissible d'un espace rigide coherent, analogue de la topologie de Zariski d'un schema. Parmi les sujets traites figurent l'etude des faisceaux coherents et de leur cohomologie, le theoreme de platification par eclatements admissibles qui generalise au cadre formel-rigide un theoreme de Raynaud-Gruson dans le cadre algebrique, et le theoreme de comparaison du type GAGA pour les faisceaux coherents. Ce volume contient aussi de larges rappels et complements de la theorie des schemas formels de Grothendieck. Ce traite est destine tout autant aux etudiants ayant une bonne connaissance de la geometrie algebrique et souhaitant apprendre la geometrie rigide qu'aux experts en geometrie algebrique et en theorie des nombres comme source de references.

Singularity theory is a far-reaching extension of maxima and minima investigations of differentiable functions, with implications for many different areas of mathematics, engineering (catastrophe theory and the theory of bifurcations), and science. The three parts of this first volume of a two-volume set deal with the stability problem for smooth mappings, critical points of smooth functions, and caustics and wave front singularities. The second volume describes the topological and algebro-geometrical aspects of the theory: monodromy, intersection forms, oscillatory integrals, asymptotics, and mixed Hodge structures of singularities.

The first volume has been adapted for the needs of non-mathematicians, presupposing a limited mathematical background and beginning at an elementary level. With this foundation, the book's sophisticated development permits readers to explore more applications than previous books on singularities."

The present volume is the second in a two-volume set entitled "Singularities of Differentiable Maps." While the first volume, subtitled "Classification of Critical Points" and originallypublishedas Volume82 in the Monographs in Mathematics series, contained the zoology of differentiable maps, that is, it was devoted to a description of what, where, and how singularities could be encountered, this second volume concentrates on elements of theanatomy and physiology of singularities of differentiable functions. The questions considered are about the structure of singularities and how they function."

The present monograph further develops the study, via the techniques of combinatorial anabelian geometry, of the profinite fundamental groups of configuration spaces associated to hyperbolic curves over algebraically closed fields of characteristic zero. The starting point of the theory of the present monograph is a combinatorial anabelian result which allows one to reduce issues concerning the anabelian geometry of configuration spaces to issues concerning the anabelian geometry of hyperbolic curves, as well as to give purely group-theoretic characterizations of the cuspidal inertia subgroups of one-dimensional subquotients of the profinite fundamental group of a configuration space. We then turn to the study of tripod synchronization, i.e., of the phenomenon that an outer automorphism of the profinite fundamental group of a log configuration space associated to a stable log curve induces the same outer automorphism on certain subquotients of such a fundamental group determined by tripods [i.e., copies of the projective line minus three points]. The theory of tripod synchronization shows that such outer automorphisms exhibit somewhat different behavior from the behavior that occurs in the case of discrete fundamental groups and, moreover, may be applied to obtain various strong results concerning profinite Dehn multi-twists. In the final portion of the monograph, we develop a theory of localizability, on the dual graph of a stable log curve, for the condition that an outer automorphism of the profinite fundamental group of the stable log curve lift to an outer automorphism of the profinite fundamental group of a corresponding log configuration space. This localizability is combined with the theory of tripod synchronization to construct a purely combinatorial analogue of the natural outer surjection from the etale fundamental group of the moduli stack of hyperbolic curves over the field of rational numbers to the absolute Galois group of the field of rational numbers.

Although much of classical ergodic theory is concerned with single transformations and one-parameter flows, the subject inherits from statistical mechanics not only its name, but also an obligation to analyze spatially extended systems with multidimensional symmetry groups. However, the wealth of concrete and natural examples which has contributed so much to the appeal and development of classical dynamics, is noticeably absent in this more general theory. The purpose of this book is to help remedy this scarcity of explicit examples by introducing a class of continuous Zd-actions diverse enough to exhibit many of the new phenomena encountered in the transition from Z to Zd, but which nevertheless lends itself to systematic study: the Zd-actions by automorphisms of compact, abelian groups. One aspect of these actions, not surprising in itself but quite striking in its extent and depth nonetheless, is the connection with commutative algebra and arithmetical algebraic geometry. The algebraic framework resulting from this connection allows the construction of examples with a variety of specified dynamical properties, and by combining algebraic and dynamical tools one obtains a quite detailed understanding of this class of Zd-actions."

After Pyatetski-Shapiro[PS1] and Satake [Sa1] introduced, independent of one another, an early form of the Jacobi Theory in 1969 (while not naming it as such), this theory was given a de?nite push by the book The Theory of Jacobi Forms by Eichler and Zagier in 1985. Now, there are some overview articles describing the developments in the theory of the Jacobigroupandits autom- phic forms, for instance by Skoruppa[Sk2], Berndt [Be5] and Kohnen [Ko]. We refertotheseformorehistoricaldetailsandmanymorenamesofauthorsactive inthistheory,whichstretchesnowfromnumbertheoryandalgebraicgeometry to theoretical physics. But let us only brie?y indicate several- sometimes very closely related - topics touched by Jacobi theory as we see it: * ?eldsofmeromorphicandrationalfunctionsontheuniversalellipticcurve resp. universal abelian variety * structure and projective embeddings of certain algebraic varieties and homogeneous spaces * correspondences between di?erent kinds of modular forms * L-functions associated to di?erent kinds of modular forms and autom- phic representations * induced representations * invariant di?erential operators * structure of Hecke algebras * determination of generalized Kac-Moody algebras and as a ? nal goal related to the here ?rst mentioned * mixed Shimura varieties and mixed motives. Now, letting completely aside the arithmetical and algebraic geometrical - proach to Jacobi forms developed and instrumentalized by Kramer [Kr], we ix x Introduction will treat here a certain representation theoretic point of view for the Jacobi theory parallel to the theory of Jacquet-Langlands [JL] for GL(2) as reported by Godement [Go2], Gelbart [Ge1] and, recently, Bump [Bu].

by a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq G] by requiring it to possess the latter as a comodule. A third principle is to focus attention on the tensor structure of the cat egory of ( ; modules. This means of course just defining an algebra structure on Rq G]; but this is to be done in a very specific manner. Concretely the category is required to be braided and this forces (9.4.2) the existence of an "R-matrix" satisfying in particular the quantum Yang-Baxter equation and from which the algebra structure of Rq G] can be written down (9.4.5). Finally there was a search for a perfectly self-dual model for Rq G] which would then be isomorphic to Uq(g). Apparently this failed; but V. G. Drinfeld found that it could be essentially made to work for the "Borel part" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra. This gives Uq(g) up to passage to a quotient. One of the most remarkable aspects of the above superficially different ap proaches is their extraordinary intercoherence. In particular they essentially all lead for G semisimple to the same and hence "canonical," objects Rq G] and Uq(g), though this epithet may as yet be premature."

This book is an introduction to two higher-categorical topics in algebraic topology and algebraic geometry relying on simplicial methods. It is based on lectures - livered at the Centre de Recerca Matem ati ca in February 2008, as part of a special year on Homotopy Theory and Higher Categories. Ieke Moerdijk's lectures constitute an introduction to the theory ofdendroidal sets, an extension of the theory of simplicial sets designed as a foundation for the homotopy theory of operads. The theory has many features analogous to the theory of simplicial sets, but it also reveals many new phenomena, thanks to the presence of automorphisms of trees. Dendroidal sets admit a closed symmetric monoidal structure related to the Boardman{Vogt tensor product. The lecture notes develop the theory very carefully, starting from scratch with the combinatorics of trees, and culminating with a model structure on the category of dendroidal sets for which the brant objects are the inner Kan dendroidal sets. The important concepts are illustrated with detailed examples.

It is now more than five years since the Belgian block cipher Rijndael was chosen as the Advanced Encryption Standard {AES). Joan Daemen and Vincent Rijmcn used algebraic techniques to provide an unparalleled level of assurance against many standard statistical cryptanalytic tech- niques. The cipher is a fitting tribute to their distinctive approach to cipher design. Since the publication of the AES, however, the very same algebraic structures have been the subject of increasing cryptanalytic attention and this monograph has been written to summarise current research. We hope that this work will be of interest to both cryptogra- phers and algebraists and will stimulate future research. During the writing of this monograph we have found reasons to thank many people. We are especially grateful to the British Engineering and Physical Sciences Research Council (EPSRC) for their funding of the research project Security Analysis of the Advanced Encryption System (Grant GR/S42637), and to Susan Lagerstrom-Fifc and Sharon Palleschi at Springer. Wo would also hke to thank Glaus Diem, Maura Paterson, and Ludovic Perret for their valuable comments. Finally, the support of our families at home and our colleagues at work has been invaluable and particularly appreciated.

Bringing together two fundamental texts from Frederic Pham's research on singular integrals, the first part of this book focuses on topological and geometrical aspects while the second explains the analytic approach. Using notions developed by J. Leray in the calculus of residues in several variables and R. Thom's isotopy theorems, Frederic Pham's foundational study of the singularities of integrals lies at the interface between analysis and algebraic geometry, culminating in the Picard-Lefschetz formulae. These mathematical structures, enriched by the work of Nilsson, are then approached using methods from the theory of differential equations and generalized from the point of view of hyperfunction theory and microlocal analysis.

Providing a 'must-have' introduction to the singularities of integrals, a number of supplementary references also offer a convenient guide to the subjects covered.

This book will appeal to both mathematicians and physicists with an interest in the area of singularities of integrals.

Frederic Pham, now retired, was Professor at the University of Nice. He has published several educational and research texts. His recent work concerns semi-classical analysis and resurgent functions."

The algorithmic solution of problems has always been one of the major concerns of mathematics. For a long time such solutions were based on an intuitive notion of algorithm. It is only in this century that metamathematical problems have led to the intensive search for a precise and sufficiently general formalization of the notions of computability and algorithm. In the 1930s, a number of quite different concepts for this purpose were pro posed, such as Turing machines, WHILE-programs, recursive functions, Markov algorithms, and Thue systems. All these concepts turned out to be equivalent, a fact summarized in Church's thesis, which says that the resulting definitions form an adequate formalization of the intuitive notion of computability. This had and continues to have an enormous effect. First of all, with these notions it has been possible to prove that various problems are algorithmically unsolvable. Among of group these undecidable problems are the halting problem, the word problem theory, the Post correspondence problem, and Hilbert's tenth problem. Secondly, concepts like Turing machines and WHILE-programs had a strong influence on the development of the first computers and programming languages. In the era of digital computers, the question of finding efficient solutions to algorithmically solvable problems has become increasingly important. In addition, the fact that some problems can be solved very efficiently, while others seem to defy all attempts to find an efficient solution, has called for a deeper under standing of the intrinsic computational difficulty of problems."

Graphs drawn on two-dimensional surfaces have always attracted researchers by their beauty and by the variety of difficult questions to which they give rise. The theory of such embedded graphs, which long seemed rather isolated, has witnessed the appearance of entirely unexpected new applications in recent decades, ranging from Galois theory to quantum gravity models, and has become a kind of a focus of a vast field of research. The book provides an accessible introduction to this new domain, including such topics as coverings of Riemann surfaces, the Galois group action on embedded graphs (Grothendieck's theory of "dessins d'enfants"), the matrix integral method, moduli spaces of curves, the topology of meromorphic functions, and combinatorial aspects of Vassiliev's knot invariants and, in an appendix by Don Zagier, the use of finite group representation theory. The presentation is concrete throughout, with numerous figures, examples (including computer calculations) and exercises, and should appeal to both graduate students and researchers.

This two-part EMS volume provides a succinct summary of complex algebraic geometry, coupled with a lucid introduction to the recent work on the interactions between the classical area of the geometry of complex algebraic curves and their Jacobian varieties. An excellent companion to the older classics on the subject.

The first contribution by Carter covers the theory of finite groups of Lie type, an important field of current mathematical research. In the second part, Platonov and Yanchevskii survey the structure of finite-dimensional division algebras, including an account of reduced K-theory.

This EMS volume provides an exposition of the structure theory of Fano varieties, i.e. algebraic varieties with an ample anticanonical divisor. This book will be very useful as a reference and research guide for researchers and graduate students in algebraic geometry.

An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some elementary topology. More than 400 exercises distributed throughout the book offer specific examples as well as more specialised topics not treated in the main text, while three appendices present brief accounts of some areas of current research. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra.
Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of "Residues and Duality," "Foundations of Projective Geometry," "Ample Subvarieties of Algebraic Varieties," and numerous research titles.

Approximate Commutative Algebra is an emerging field of research which endeavours to bridge the gap between traditional exact Computational Commutative Algebra and approximate numerical computation. The last 50 years have seen enormous progress in the realm of exact Computational Commutative Algebra, and given the importance of polynomials in scientific modelling, it is very natural to want to extend these ideas to handle approximate, empirical data deriving from physical measurements of phenomena in the real world. In this volume nine contributions from established researchers describe various approaches to tackling a variety of problems arising in Approximate Commutative Algebra.

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.

This is a description of the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today. It is intended for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites. Few algebraic prerequisites are presumed beyond a basic course in linear algebra.

This monograph is a self-contained introduction to the geometry of Riemann Surfaces of constant curvature 1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces. Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations.

This book is the result of a 25-year-old project and comprises a collection of more than 500 attractive open problems in the field. The largely self-contained chapters provide a broad overview of discrete geometry, along with historical details and the most important partial results related to these problems. This book is intended as a source book for both professional mathematicians and graduate students who love beautiful mathematical questions, are willing to spend sleepless nights thinking about them, and who would like to get involved in mathematical research.

In this book, the author writes freely and often humorously about his life, beginning with his earliest childhood days. He describes his survival of American bombing raids when he was a teenager in Japan, his emergence as a researcher in a post-war university system that was seriously deficient, and his life as a mature mathematician in Princeton and in the international academic community. Every page of this memoir contains personal observations and striking stories. Such luminaries as Chevalley, Oppenheimer, Siegel, and Weil figure prominently in its anecdotes.

Goro Shimura is Professor Emeritus of Mathematics at Princeton University. In 1996, he received the Leroy P. Steele Prize for Lifetime Achievement from the American Mathematical Society. He is the author of Elementary Dirichlet Series and Modular Forms (Springer 2007), Arithmeticity in the Theory of Automorphic Forms (AMS 2000), and Introduction to the Arithmetic Theory of Automorphic Functions (Princeton University Press 1971)."

This edition has been called startlingly up-to-date, and in this corrected second printing you can be sure that it 's even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories.

Absolute values and their completions -like the p-adic number fields- play an important role in number theory. Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible. In valuation theory, the notion of a completion has to be replaced by that of the so-called Henselization.

In this book, the theory of valuations as well as of Henselizations is developed. The presentation is based on the knowledge acquired in a standard graduate course in algebra. The last chapter presents three applications of the general theory -for instance to Artin's Conjecture on the p-adic number fields- that could not be obtained by the use of absolute values alone.

This book gives an account of theoretical and algorithmic developments on the integral closure of algebraic structures. It gives a comprehensive treatment of Rees algebras and multiplicity theory while pointing to applications in many other problem areas. Its main goal is to provide complexity estimates by tracking numerically invariants of the structures that may occur.